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How to use Euler's angle theorem in rotation of a coordinate 
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#1
Dec412, 07:49 PM

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If i have a point at (0,0,5) in x,y,z system, then i make 2 rotation on the point with center at origin.
i)the first rotation is on yaxis with angle P in clockwise direction. ii) the second rotation is on the point's new xaxis rotate in angle Q in clockwise direction. How can i find the new coordinates after these 2 rotation with Euler's rotation theorem? 


#2
Dec412, 10:38 PM

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P: 9,930

That theorem only tells you the net transformation is a rotation. Rodrigues' rotation formula might be more use.



#3
Dec512, 02:57 AM

P: 9

using the point (0,0,5) , i make a rotation of 30 degree in clockwise direction about y axis,
so i should get the rotation matrix of  cos 30 0 sin 30  x  cos 30 0 sin 30  x  0 1 0  y or  0 1 0  y  sin 30 0 cos 30 z  sin 30 0 cos 30 z which one is the right one or i got both wrong? 


#4
Dec512, 05:59 AM

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How to use Euler's angle theorem in rotation of a coordinate
Second one is correct (I think, they don't line up very well).
One rotation matrix for positive (direction of a right hand screw, or counterclockwise as viewed from the outside) rotations:
For negative (or clockwise), reverse the sign of the sines. Since you're dealing with orthogonal matrices, the safer way is that clockwise is the inverse of counterclockwise and the inverse of an orthogonal right hand matrix (which your rotation matrices are) is the transpose. In other words, take the transpose of the counterclockwise matrix to get the clockwise rotation matrix. The latter helps you when you start combining rotations into a single matrix and you don't want to have to reaccomplish the work for the opposite direction. Additionally, sometimes the known angle won't be the rotation angle, but the complement of the rotation angle. Instead of building an ugly rotation matrix consisting of cos (90  x), sine will substituted for cosine and cosine for sine to make a cleaner matrix consisting of just x. 


#5
Dec612, 02:11 AM

P: 9

lets say if i have a Zaxis which perpendicularly point into the screen, Xaxis which point to its right side and Yaxis which point upward and a point with coordinate ( 0 , 0 , 5 ).
then i rotate the point's yaxis 30 degree in clockwise direction. so the rotation operation matrix should be
then on the point's new xaxis, i rotate the point 60 degree counter clockwise, so the point will move upward. And the matrix is:



#6
Dec612, 07:05 PM

P: 9

What is the A=BCD mean in this page, http://mathworld.wolfram.com/EulerAngles.html and how do they get the value of a11 , a12 , a13...a33, i thought it is just multiplication of matrices



#7
Dec712, 01:37 AM

P: 9

I get it now, thanks everyone for teaching me.



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